# Article

# Article

- Mathematics
- Geometry
- Ellipse

# Ellipse

Article By:

**Blumenthal, Leonard M. **Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.

Last reviewed:June 2020

DOI:https://doi.org/10.1036/1097-8542.228700

**A member of the class of curves that are intersections of a plane with a cone of revolution.** The ellipse is obtained when the plane cuts all the elements of one nappe, and does not go through the apex. In the **illustration**, denote the distance between two points, *F*, *F*′ of a plane by 2*c*, *c* > 0, and let 2*a* be a constant, with *a* > *c*. The ellipse with foci *F* and *F*′ and major axis 2*a* is the locus of points *P* of the plane such that *PF* + *PF*′ = 2*a*, where *PF* denotes the distance of *P* and *F*. This suggests the following construction of an ellipse. Put pins at *F* and *F*′, and slip over them a loop of thread of length 2*a* + 2*c*, pulling the thread taut with a pencil. If the pencil is moved, keeping the thread taut, its point traces an ellipse. Another way to construct an ellipse is to drill a hole in a stick (at any point other than the midpoint) and move the stick so that its ends slide along two mutually perpendicular lines. The point of a pencil inserted in the hole will trace an ellipse. Limiting forms of the ellipse are (1) a circle, as the two foci approach coincidence, and (2) the segment *FF*′, as *c* approaches *a*. If a circle is projected orthogonally on a plane not parallel to the plane of the circle, an ellipse is obtained, and every ellipse may be so obtained. Lines joining the foci to a point *P* of an ellipse make equal angles with the tangent to the ellipse at *P*, and consequently light or sound that emanates from one focus is reflected to the other focus. This property is used in construction of “whispering galleries.”* See also: ***Conic section**

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